Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:

(1) What is the maximum of the set $\{d_i/d_{i-1}: 1\leq i \leq l\}$. Say it $M$.

Assume that you know the prime factorization of $n (= p_1^{\alpha_1}..p_r^{\alpha_r})$. Can I have a formula in terms of $p_i$'s and $\alpha_i$'s. One of the crude upper bound can be $p_r$ but this is really bad if $n$ has many distinct prime factors.

(2) Instead of maximum, the average may be more interesting and useful. So can we estimate
the mean and variance,

But $H(x, y, z) - H(x-\Delta, y, z)$ can be computed only if $\Delta \geq x/\log^{10}{z}$, (as in Introduction, Theorem 2, in above mentioned paper). Then recovering the integer whose divisor we have to look at is not possible by this formula.
–
KamalakshyaDec 7 '12 at 16:37

This may be worth studying if n has many divisors. What significance do you see it having for n having fewer than 5 divisors? Or 10? Gerhard "Ask Me About System Design" Paseman, 2012.12.07
–
Gerhard PasemanDec 7 '12 at 17:11

Also, assuming d_0 is 1, it is clear that M is at least p_1, where I assume the prime factors are indexed in order of increasing magnitude, and that for n with many factors, log of the mean will be close to if not equal to log of n divided by number of divisors of n. There may be some bizarre exceptions like a twice a large ppprime taken to a large power, but they should be easy to characterize. Gerhard "Ask Me About Gut Feeling" Paseman, 2012.12.07
–
Gerhard PasemanDec 7 '12 at 17:21